A key condition in the analysis above is equation (REF ), which implies (REF )
for the eigenbundles of \(K\) .
For real \(K\) , this means that the signature of \(\eta \)  must be \((d,d)\) ; so in four dimensions this is \((++--)\) ,
which is different from the usual Lorentz signature \((+---)\)  of general relativity.
However, if we allow \(K\)  to be a map on the complexified tangent bundle,
then the metric is allowed to have any signature (or to be complex).
Our interest in complex-valued maps arises from the power of using complex methods in relativity,
which date back to remarkable programmes such as Penrose's twistor theory, Newman's H-space,
and Plebański's hyper-heavenly (HH) construction.
(One can argue that, in a sense, all these constructions can be understood as parts of the twistor programme.)
This is one of our main motivations in this work:
submanifolds where the condition (REF ) holds are actually the basic object of twistor theory,
but they arise from a completely different perspective, namely the essential premise of the twistor programme that light-rays
are more fundamental than spacetime points.
A related motivation comes from the fact that
a “complex-valued almost-complex structure” has some significance in developments in mathematical relativity
and their relation to twistor structures, see [1]}, [2]}, [3]}, [4]}, [5]}.
As in ordinary para-Hermitian geometry, the phenomenon of “half-integrability” is shared by such complex-valued maps.
The present work can be regarded as the application of some of the modern DFT-like techniques mentioned above
to 4-manifolds, with the extra ingredient that \(K\)  can be complex-valued so that one can make contact with
general relativity and twistor theory.
